Two Results on the Complex Oscillation Theory of Differential Equations with Polynomial Coefficients

Abstract:

      In this paper, we prove: if a_ｋ－ｊ(j=1,…,k) are polynomials, deg a_ｋ－ｊ=n_ｋ－ｊ, there exists some a_ｋ－s(1≤s≤k) such that n_ｋ－ｊ/j≤n_ｋ－s/s if 1≤jｋ－ｊ< n_ｋ－s- ( j - s) if s < j ≤ k , if F≠0 is an entire function satisfying σ（Ｆ）＝β＜（ｎ_ｋ－s＋ｓ）／ｓ, then a so lution of the differential equat ion ｆ^（ｋ）＋ａ_ｋ－1ｆ^（ｋ-1）＋…＋ａ₀ｆ＝Ｆsat isf ies λ（ｆ）＝λ（ｆ）＝σ（ｆ）＝（ｎ_ｋ－s＋ｓ）／ｓ，or σ（ｆ）＝β.